pubs.acs.org/Macromolecules Published on Web 06/05/2009 r 2009 American Chemical Society

4746 Macromolecules 2009, 42, 4746–4750

DOI: 10.1021/ma900199t

Branch Content of Metallocene Polyethylene

Ramnath Ramachandran, Gregory Beaucage,* and Amit S. Kulkarni

Department of Chemical and Materials Engineering, University of Cincinnati, Cincinnati, Ohio 45221

Douglas McFaddin, Jean Merrick-Mack, and Vassilios Galiatsatos

Equistar Chemicals, LP, a LyondellBasell Company, Cincinnati Technology Center, 11530 Northlake Drive,Cincinnati, Ohio 45249

Received January 29, 2009; Revised Manuscript Received May 14, 2009

ABSTRACT: Small-angle neutron scattering (SANS) is employed to investigate the structure and long-chain branch (LCB) content of metallocene-catalyzed polyethylene (PE). A novel scaling approach is appliedto SANS data to determine themole fraction branch content (φbr) of LCBs in PE. The approach also providesthe average number of branch sites per chain (nbr) and the average number of branch sites per minimum path(nbr,p). These results yield the average branch length (zbr) and number of inner segments ni, giving furtherinsight into the chain architecture. The approach elucidates the relationship between the structure andrheological properties of branched polymers. This SANS method is the sole analytic measure of branch-on-branch structure and average branch length for topologically complex macromolecules.

Introduction

Structural branching is known tooccur in a varietyofmaterialssuch as polymers and ceramic aggregates.1,2 Owing to theinfluence of branching on the physical and chemical propertiesof these materials, a universal technique to quantify the branchcontent has long been sought. Branching in polymers can beclassified into short-chain branching (SCB), referring to branchesthat have only a few carbon atoms and aremuch smaller than thebackbone to which they are attached, and long-chain branching(LCB) where the length of the branch is comparable to that of thebackbone. The effects of short-chain branching have been dis-cussed previously.3 A new scaling model4 has recently beendeveloped to quantify long-chain branch content in such ramifiedstructures. The scaling model has been employed successfully tostudy ceramic aggregates4 and to describe the folded and un-folded state in proteins and RNA.5 This model can be used toquantify long-chain branching in polymers as well. Long-chainbranching has significant influence on the physical propertiesdisplayed by a polymer. The presence of long-chain branches(LCBs) considerably affects the structure and consequently therheological properties and processability of polymers.6-9 Hence,an efficient and comprehensive method to quantify long-chainbranch (LCB) content in polymers has been desired for manyyears. Various techniques based on size-exclusion chromatogra-phy (SEC),10 nuclear magnetic resonance (NMR) spectro-scopy,11 and rheology12,13 have been utilized to determine LCBcontent in branched polymers. The drawbacks of these existingtechniques have been previously discussed.14 SEC is ineffective incharacterizing low levels of LCB. NMR cannot distinguishbetween short- and long-chain branches, and rheological esti-mates of LCB, though sensitive to low levels of LCB, tend to besemiempirical and qualitative in nature.14 However, NMR is aneffective technique to determine the total number of branch sites(β) in some polymer chains. In this paper, a scaling model4 isapplied to small-angle neutron scattering (SANS) data obtained

from dilute solutions of metallocene polyethylene samples toquantify the LCB content in polymers. These samples have beenpreviously studied in the literature.12,13 New data provided fromthis model will allow polymer chemists to better understandrheological consequences of polymer structures while developingnew catalysts for complex polymer architectures.

A polyethylene (PE) chain can be considered to exhibit twostructural levels: the overall radius of gyration Rg with massfractal dimension df and the substructural rodlike persistencelength lp orKuhn length lk=2lp.

15These features canbeobservedin a small-angle scatteringpattern and can be determined throughthe application of local scattering laws and mass-fractal powerlaws under the Rayleigh-Gans approximation. Local scatteringlaws such as Guinier’s law and power laws describe these levels.Guinier’s law is given by16

IðqÞ ¼ G exp-q2Rg

2

3

!ð1Þ

where I(q) is the scattered intensity, scattering vector q= 4π sin(θ/2)/λ, θ is the scattering angle, λ is the wavelength of radiation,Rg is the coil radius of gyration, andG is defined asNpnp

2, whereNp is the number density of polymer coils, and np the squareof the difference in neutron scattering length density betweencoil and solvent times the square of the volume of the coil. Themass-fractal power law is another local scattering law.

IðqÞ ¼ Bfq-df for 1edf < 3 ð2Þ

It describes a mass-fractal object of dimension df, where Bf is thepower law prefactor. Together, they give an account of localfeatures like size (Rg and lp) and surface/mass scaling.4,17,18 Theunified function developed by Beaucage,3,17-19 used here, isuseful to examine small-angle scattering data from fractal struc-tures with multiple structural levels such as carbon nanotubes20

and polyethylene.*Corresponding author.

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Article Macromolecules, Vol. 42, No. 13, 2009 4747

Beaucage4 has described a scaling model which can be em-ployed to quantify branch content in polymers. A branchedpolymer chain of size Rg,2 is considered to be composed of zfreely jointedKuhn steps each of size lk

3,19 as shown in Figure 1a.The structure of the branched polymer can further be decom-

posed into an averageminimum path p (number of Kuhn steps inthe minimum path) through the structure as shown in Figure 1a.p is an average traversing path through the chain structure.A scaling relationship can be proposed between z and p4

z ¼ pc ¼ sdmin ð3Þwhere c is the connectivity dimension assuming the scalingprefactor to be one. The minimum path p is a mass-fractal ofdimension dmin and size r∼ p1/dmin while the total chain of zKuhnsteps has a dimension df g dmin and the same size r ∼ z1/df. Aparameter s (eq 3) can also be defined using dmin that reflects thenumber of steps required to connect all branch points and endpoints in the polymer structure by straight lines (size r ∼ s1/c).Substituting p ∼ rdmin in eq 3 and comparing with z ∼ rdf yieldsdf = cdmin, which shows that the chain scaling (df, z) can bedecomposed into contributions from chain tortuosity (dmin, p)and chain connectivity (c, s). For a linear polymer chain dmin= dfand c= 1. On the other hand, for a completely branched objectlike a sphere or a disk, where a linear minimum path can betraversed, df = c and dmin = 1. The minimum path dimension,dmin, and connectivity dimension c represent different features ofthe branched chain.While c increaseswith increasedbranching orconnectivity, dmin increases with tortuosity in the chain, driven bythe thermodynamics in a dilute polymer solution. That is, for alinear chain in good solvent dmin=5/3 and c=1.For a branchedchain, 1 < c e df and 1 e dmin e 5/3. dmin deviates from 5/3because the minimum path can find shortcuts through thebranched structure. From the scaling model, the mole fractionbranches (φbr) is given by4

φbr ¼ z-p

z¼ 1-zð1=cÞ-1 ð4Þ

Equations 1 and 2 can be used to calculate df, c, p, and s. dmin canbe calculated from3,4

dmin ¼ BfRdfg, 2

CpΓdf2

� �G

ð5Þ

where Cp is the polydispersity factor3,21 and Γ is the gammafunction. The quantities in eqs 4 and 5 can be acquired from

small-angle neutron scattering (SANS) data from dilute hydro-genous polymer solutions in deuterated solvent fit to the unifiedfunction which is given by3,17,18

IðqÞ ¼ fG2e-ðq2Rg2

2Þ=3 þ B2e-ðq2Rg1

2Þ=3ðq�2Þ-df2g

þ fG1e-ðq2Rg1

2Þ=3 þ B1ðq�1Þ-1g ð6Þwhere qi* = [q/{erf(qkRgi/

√6)}3] and k ≈ 1.06. The terms in the

first bracket with subscript “2” represent the good solvent scalingregime with mass-fractal dimension df2 ≈ 5/3 and the secondbracket with subscript “1” represent the rodlike persistence scalingregime. In each set of brackets, the first term gives the Guinierexponential decay and the second term yields the structurallylimited power law. The SANS data, as shown in Figure 2,contains four distinguishable features, each providing two valuesassociated with the q and I(q) positions of these features. Atlowest qweobserve the plateau I(q) value,G2, andRg2 for the coil.In the scaling regime, the slope and prefactor for the power-lawdecay are observed. Near the persistence length, a transition inslope is seen with corresponding I(q) and q values. At high q apower-law decay of -1 with a power-law prefactor is observed.These eight observable features aremodeled with a six-parameter

Figure 1. (a, left) Schematic of a branched polymer. The polymer is composed ofKuhn steps of length lk. The dark lines represent an averageminimumpath p of dimension dmin. The lighter lines represent the long-chain branches. (b, right) Connective path represented by straight lines connecting branchpoints and free ends (gray dots), of total size s and connectivity dimension c.

Figure 2. SANS data for sample HDB-1 fit to the unified fit.17,18 Datashown above were obtained at IPNS.

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4748 Macromolecules, Vol. 42, No. 13, 2009 Ramachandran et al.

function where the variables that are floated while executingthe unified fit areCp, df,Rg2, B2, dmin, and B1. All the parametersused in the unified function have been described in detailpreviously.3,19

For a long-chain branched polymer, as shown Figure 1a,branch sites occur along the minimum path through the struc-ture. The average minimum path is composed of segmentsbetween branch points or chain ends having an average numberof Kuhn steps, (p/ns,p), where ns,p is the average number ofsegments per minimum path. The end-to-end distance of theminimum path, r, in units of number of Kuhn steps can then begiven by

r ¼ ns, pp

ns, p

!3=5

ð7Þ

For a branched polymer chain, r can also be described in termsof p (Figure 1a) as

r ¼ p1=dmin ð8ÞEquating eqs 7 and 8, we obtain the following relationship:

ns, p ¼ ½pð1=dminÞ-ð3=5Þ�5=2 ð9ÞThe number of branch sites per minimum path is then given by

nbr;p ¼ ns;p -1 ð10ÞThe average number of Kuhn steps in a segment, nk,s, can bedescribed by

nk;s ¼ p

nbr;p þ 1¼ z

2nbr þ 1ð11Þ

where nbr is the number of branch sites per chain, nbr,p+1 is thenumber of segments in theminimumpath, and 2nbr+1 is the totalnumber of segments in the polymer.

From eq 3, we can rewrite eq 11 as

nbr ¼ z½ð5=2df Þ-ð3=2cÞ� þ½1-ð1=cÞ� -1

2ð12Þ

This quantity is equivalent to the average number of branches perchain, β, obtained from NMR.4,14,22

The mole fraction branch content, φbr, combined with nbr canbe used to estimate a new quantity, the weight-average branchlength (zbr), from the following relationship

zbr ¼ zφbrMKuhn

nbrð13Þ

whereMKuhn is themass of oneKuhn step as determined from theKuhn length, lk = 2lp, of a polyethylene sample,3 which is lk �13.4 g/(mol A). The quantity nbr obtained from this analysis iscompared with β obtained from NMR.22

Materials and Methods

We used metallocene-catalyzed model branched polyethylenechains with low degrees of structural branching and narrowmolecular weight distributions (Table 1). These samples havebeen extensively studied and characterized in the literature.12,13,22

SANS was performed on dilute solutions of these model poly-ethylenes in deuterated p-xylene, which is a good solvent forpolyethylene at 125 �C.Deuterated p-xylene was purchased fromSigma-Aldrich. The samples were equilibrated at 125 �C for 3 hprior to the measurements to ensure complete dissolution of thesolute. 1 wt % solutions were used, well below the overlapconcentration as described by Murase et al.23 as verified bysuperposition of concentration normalized data from 0.25, 0.5,and 1 wt % for the same polymer solution. SANS experimentswere carried out at the Intense Pulsed Neutron Source (IPNS),Argonne National Laboratory, Argonne, and NG-7 SANS24 atNational Institute of Standards and Technology (NIST) Centerfor Neutron Research (NCNR), Gaithersburg. Standard datacorrection procedures for transmission and incoherent scatteringalong with secondary standards were used to obtain I(q) vs q inabsolute units.25 Experimental runs took approximately 4 h persample at IPNS and 2 h at NIST.

Results and Discussion

Corrected SANS data are plotted in a log-log plot of I(q) vsq, as shown in Figure 2. The data are fit to the unified function(eq 6)3,17-19 followed by the application of the scaling model.4

The dotted curve in Figure 2 represents the Guinier exponentialdecay at low q. The dashed curve in Figure 2 represents theunified fit at the persistence substructural level. As long asreasonable starting values are chosen, for each sample analysis,the unified fit was robust with rapid convergence to the samevalues. Reasonable starting values can be determined by visuallyinspecting the four main features of the scattering curve, de-scribed above. Table 1 lists the sample names, NMR branchcontent in terms of number of long-chain branches per 1000carbon atoms,22 LCB/103C, number-average molecular weight22

Mn, polydispersity index (Mw/Mn),22 PDI, and average number

of branch sites per chain, β, from NMR.22 The NMR branchcontent, in terms of number of long-chain branches per 1000carbon atoms, is converted to average number of branch sites perchain, β, using the relationship22

β ¼ ðno: of LCB=1000ÞNMRMn

14000ð14Þ

where Mn is the number-average molecular weight of the poly-ethylene and 14000 g/mol refers to the molar mass of 1000backbone carbons. Table 1 further lists the quantities calculatedfrom SANS including number of branch sites per chain from eqs

Table 1. Characterization of Long-Chain Branching in Dow HDB Samples

sample LCB/103C 13C NMRa Mn (g/mol)a PDI (Mw/Mn)a β nbr nbr,p φbr zbr (g/mol)

HDB-1 0.026 39 300 1.98 0.073 0.080( 0.004 0.047( 0.005 0.10( 0.02 12 700( 1500HDB-2 0.037 41 500 1.93 0.110 0.115( 0.005 0.053( 0.005 0.14( 0.02 17 400( 1600HDB-3 0.042 41 200 1.99 0.124 0.144( 0.007 0.065( 0.005 0.17( 0.02 16 500( 1600HDB-4 0.080 39 200 2.14 0.224 0.262( 0.007 0.090( 0.008 0.28( 0.03 18 600( 1700

aReference 22.

Table 2. Size and Dimensions of Dow HDB Samples Measuredfrom SANS

sample Rg (A) df c lp (A)

HDB-1 95( 6 1.70( 0.02 1.03( 0.01 6.5( 0.5HDB-2 103( 8 1.71( 0.02 1.04( 0.02 6.7( 0.4HDB-3 104( 8 1.73 ( 0.02 1.05( 0.02 6.6( 0.5HDB-4 79( 4 1.78( 0.04 1.08( 0.03 6.9( 0.5

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Article Macromolecules, Vol. 42, No. 13, 2009 4749

9-12, nbr, number of branch sites per minimum path from eqs 9and 10, nbr,p, mole fraction branches from eq 4, φbr, and averagebranch length from eq 13, zbr. Table 2 lists the average radius ofgyration of the coil (Rg), mass-fractal dimension, df, connectivitydimension, c, and persistence length,3 lp, measured from SANS.Errors reported were propagated from the counting error in theraw data.

Figure 3a plots nbr calculated from eq 12 against β from ref 22with good agreement, although the value of nbr is slightly higherthan β. Figure 3b plots nbr,p calculated from eq 10 against nbr fromeq 12.While nbr measures every branch point in a polymer chain,nbr,p reflects the number of branch points in the minimum path.For a comb or 3-arm star structure (top inset, Figure 3b), withno branch-on-branch structure, it is expected that nbr,p=nbr.For a more complex structure displaying branch-on-branchtopology, nbr,p<nbr. Although we observe a monotonic relation-ship in the plot, nbr,p is lower than nbr for the HDB samples. Thisimplies the presence of branch-on-branch architecture inthese samples. The plot of nbr,p vs nbr in Figure 3b shows astronger deviation at higher branch content. The nbr,p valueplateaus at about 0.1 while nbr increases to almost 0.3.The plateau in nbr,p indicates that the minimum path branchcontent reaches a constant value while additional branches areadded through branch-on-branch structures. The system consists

of a few hyperbranched chains in a majority of linear chains athigh branch content rather than a uniform distribution ofbranching because nbr is less than 1 and due to the noteddifference between nbr,p and nbr.

As mentioned earlier, the average branch length, zbr, can havea significant impact on understanding the rheological behavior oflong-chain branched polymers. Figure 4a shows a log-linear plotof the zero shear rate viscosity enhancement, η0/η0,L, as reportedby Costeux et al.,22 versus the average branch length, zbr. Anexponential increase in the viscosity enhancementwith increasingbranch length is observed.

The extrapolation of the fit of the data in Figure 4a interceptsthe zbr axis around 9000 g/mol. This implies that the viscosityenhancement effect due to long-chain branching starts to occur,when the weight-average branch length becomes 9000 g/mol. Formodel samples with monodisperse branches,26 it has been pre-viously reported that significant rheological affects are onlyobserved for branches 2.4 times the entanglement mole-cular weight, Me, of 1250 g/mol.13 Since zbr is a weight-averagebranch length, the ratio of zbr to the literature value26 suggeststhat the higher value in the current analysis can be attributed tothe polydispersity in the branch lengths of theHDB samples. Thisimplies a polydispersity index ofMw/Mn≈ 9000/3000=3 for thebranches.

Figure 3. (a, left) Plot of number of branch sites per chain, nbr, calculated from eq 12 against average number of branches per chain, β, fromNMR.22

The dashed line represents nbr= β. (b, right) Plot of number of branch sites per minimum path, nbr,p, calculated from eq 10 against nbr calculated fromeq 12. The dotted line represents nbr = nbr,p.

Figure 4. (a, left) Log-linear plot of zero shear rate viscosity enhancement for linear polyethylene of the sameMw22 against the average length of a

branch, zbr (from SANS), for the HDB samples. The extrapolation of zbr data intercepts the zbr axis at 9000 g/mol. (b, right) Linear plot of zeroshear rate viscosity enhancement for linear polyethylene of sameMw

22 against number of inner segments per chain, ni, for theHDB samples. The errorin ni for HDB-1 is smaller than the point size.D

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4750 Macromolecules, Vol. 42, No. 13, 2009 Ramachandran et al.

On the basis of previous studies,22,27 the viscosity enhancementcan also be attributed to the number of inner segments per chain,ni, as described in ref 22, in branch-on-branch polymer chains. nican be approximated by

ni ¼ nbr -nbr;p ð15ÞFigure 4b shows a plot of viscosity enhancement, η0/η0,L, vs nicalculated in this way. The plot displays a linear increase inviscosity enhancement with increasing number of inner segmentsfor the samples studied. If the inner segments are considered notto contribute to the viscous response of the melt and since themole fraction of inner segments in the melt is proportional to ni,we can consider the polyethylene melt as a dilute solution of niinner segments per chain in a melt of mostly linear polyethylene.The linear functionality of η0/η0,L in ni is then analogous to adilute suspension of particulates. We can use the Einsteinapproximation28 for a dilute particulate suspension, η0= η0,L(1+ni[η0]) as an approximation, except that there is a shift ofabout 0.015 on the ni axis. Einstein behavior indicates that innersegments do not contribute to flow, acting as particulate inclu-sions, and showa simple volumetric exclusion from the remainingviscous material composed of linear chains and non-hyper-branched chains. In the Einstein linear equation the slope, 278chains/inner segment, corresponds to a type of intrinsic viscosityfor hyperbranched structures in a suspension of linear chains.Theshift factor of 0.015 inner segments/chain may arise due to thepresence of a small population of branched segments that are notlong enough to significantly affect viscosity. These are reflected inFigure 4a by the nonzero intercept on the zbr axis.

Conclusion

A novel scaling approach to characterize long-chain branchcontent in polyethylene has been presented. Through this ap-proach, the new quantities mole fraction branch content, φbr,number of branch sites per chain, nbr, number of branch sites perminimumpath, nbr,p, number of inner segments per chain, ni, andaverage branch length, zbr, are reported. While φbr quantifies themole fraction long-chain branch content zbr can provide addi-tional information about the architectural makeup of a polymer,resulting in improved understanding of its rheological properties.nbr,p when combinedwith nbr gives further details about the chainarchitecture. This information will give polymer and catalystchemists enhanced information to understand polymer architec-tures with desirable rheological properties. The approach en-compasses both qualitative and quantitative analysis of long-chain branching in polyethylene. The scaling model has beensuccessfully employed previously to determine branching inceramic aggregates4 and to quantify the degree of folding inproteins and RNA.5

The procedure described in this paper does have certainlimitations. First, the system being studied would need to showmass-fractal scaling so it may not be applicable to some denselybranched dendritic macromolecules for example. The method isalso not able to give a firm value for the length at which a branchbecomes a long-chain branch since there may be an overlapregime where both persistence effects associated with SCB (asdiscussed in ref 3) and scaling effects associated with LCB areobserved. The effect of polydispersity in polymers and blendingof different systems can be accommodated in this approachand will be discussed in forthcoming papers. The new scaling

approachdiscussed herewill be further testedwithmodel star andcomb polyethylene samples.

Acknowledgment. We thank L. J. Effler and A. W. deGrootof The Dow Chemical Company for providing the polyethylenesamples. This work utilized facilities supported in part by theNational Science Foundation under Agreement DMR-0454672.We acknowledge the support of the National Institute of Stan-dards and Technology (NIST), U.S. Department of Commerce,for providing the neutron research facilities used in this work.Weare grateful toB.Hammouda, S.Kline, andM.Laver atNIST fortheir valuable support and guidance during beam time at NIST.We also acknowledge the support of Intense Pulsed NeutronSource (IPNS) and thank P. Thiyagarajan for valuable supportduring preliminary work. We acknowledge the support of Equis-tar Chemicals, LP, for funding of this work.

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